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Quantization and the tangent groupoid

N. P. Landsman
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Abstract

This is a survey of the relationship between C*-algebraic deformation quantization and the tangent groupoid in noncommutative geometry, emphasizing the role of index theory. We first explain how C*-algebraic versions of deformation quantization are related to the bivariant E-theory of Connes and Higson. With this background, we review how Weyl--Moyal quantization may be described using the tangent groupoid. Subsequently, we explain how the Baum--Connes analytic assembly map in E-theory may be seen as an equivariant version of Weyl--Moyal quantization. Finally, we expose Connes's tangent groupoid proof of the Atiyah--Singer index theorem

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N. P. Landsman. 2002-10-16. Quantization and the tangent groupoid. https://arxiv.org/abs/math-ph/0208004

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